Generally no that doesn't happen. It may be very specific to Amazon though. Android uses signing keys to validate updates, and if the signing key doesn't match, it is impossible to install the update. My guess is Amazon (was?) using the same signing keys between the two stores.

Most developers use Google's app signing service to ensure that a loss of the signing key will not strand their users on old versions. In that case, it would not have been possible for Amazon distributed apps to use the same signing keys. I say would not have, because these new requirements mean it will actually now be possible since Amazon could distribute the updates published to Google Play, and doing resigning shenanigans would throw the baby out with the bathwater (allowing users to seamlessly switch app stores)